The present seminar is devoted to the theory of the so-called generalized curvatures originally appeared in the framework of the optimal control theory. Generalized curvatures adopt the concept of the Riemann curvature tensor to the integral curves of a large class of dynamical systems and provide an effective tool for their qualitative analysis.

In this seminar we will focus on certain results applicable to the Hamiltonian systems. In particular, we will discuss the role of the first integrals of a system in the structure of its curvature tensor, and the hyperbolic properties of the Hamiltonian systems of negative curvature.